You initially have 4 characters available for Story mode but if you complete the Story mode with a character and you will unlock story mode for more characters.

Unlockable	How to unlock
Elizabeth's story	Fully clear Story Mode with one of the Persona 3 characters and one of the Persona 4 characters.
Naoto, Mitsuru, Akihiko, and Aigis	Reach a cliffhanger ending with two characters in Story Mode.
Teddie and Kanji's Story	Reach a cliffhanger ending in Story Mode.
Unlock Labrys' Story	Reach the cliffhanger ending with one of the Persona 3 characters.
Unlock the full story for Mitsuru, Akihiko, AIgis, and Naoto.	Complete the full story for Yosuke, Yu, Yukiko, Chie, Kanji, or Teddie.
Unlock the full story for Yosuke, Yu, Teddie, Chie, Yukiko, and Kanji.	Complete Story Mode with Labrys.